Quotients of the Bruhat-Tits tree by arithmetic subgroups of special unitary groups

TL;DR

This paper investigates the action of arithmetic subgroups of special unitary groups on their Bruhat-Tits trees over function fields, revealing a 'spider-like' quotient structure and analyzing their algebraic and homological properties.

Contribution

It provides a detailed geometric description of the quotient graph for these groups, extending previous work and applying it to finite fields for more precise results.

Findings

The quotient graph resembles a spider with cuspidal rays attached to a connected core.

The arithmetic subgroups can be described as amalgamated products.

The core is finite when the base field is finite, enabling refined analysis.

Abstract

Let $K$ be the function field of a curve $C$ over a field $\mathbb{F}$ of either odd or zero characteristic. Following the work by Serre and Mason on $\mathrm{SL}_2$, we study the action of arithmetic subgroups of $\mathrm{SU}(3)$ on its corresponding Bruhat-Tits tree associated to a suitable completion of $K$. More precisely, we prove that the quotient graph "looks like a spider", in the sense that it is the union of a set of cuspidal rays (the "legs"), parametrized by an explicit Picard group, that are attached to a connected graph (the "body"). We use this description in order to describe these arithmetic subgroups as amalgamated products and study their homology. In the case where $\mathbb{F}$ is a finite field, we use a result by Bux, K\"ohl and Witzel in order to prove that the "body" is a finite graph, which allows us to get even more precise applications.